Solvable Subgroups and their Lie Algebras in Characteristic p

1979 ◽  
Vol 31 (2) ◽  
pp. 308-311
Author(s):  
David J. Winter
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Maximal Torus ◽  
Borel Subgroup ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Connected Component ◽  
Characteristic P ◽  
Maximal Tori ◽  
Commutative Subalgebras

1. Introduction. Throughout this paper, G is a connected linear algebraic group over an algebraically closed field whose characteristic is denoted p. For any closed subgroup H of G, denotes the Lie algebra of H and H0 denotes the connected component of the identity of H.A Borel subalgebra of is the Lie algebra of some Borel subgroup B of G. A maximal torus of is the Lie algebra of some maximal torus T of G. In [4], it is shown that the maximal tori of are the maximal commutative subalgebras of consisting of semisimple elements, and the question was raised in § 14.3 as to whether the set of Borel subalgebras of is the set of maximal triangulable subalgebras of .

scholarly journals On Commuting Varieties of Nilradicals of Borel Subalgebras of Reductive Lie Algebras

2014 ◽  
Vol 58 (1) ◽  
pp. 169-181 ◽  
Author(s):  
Simon M. Goodwin ◽  
Gerhard Röhrle
Keyword(s):  
Finite Number ◽  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Borel Subgroup ◽  
Closed Field ◽  
Reductive Algebraic Group ◽  
Algebraically Closed Field ◽  
Commuting Variety ◽  
Irreducible Components

AbstractLet G be a connected reductive algebraic group defined over an algebraically closed field of characteristic 0. We consider the commuting variety of the nilradical of the Lie algebra of a Borel subgroup B of G. In case B acts on with only a finite number of orbits, we verify that is equidimensional and that the irreducible components are in correspondence with the distinguishedB-orbits in . We observe that in general is not equidimensional, and determine the irreducible components of in the minimal cases where there are infinitely many B-orbits in .

Representations Subduced on an Ideal of a Lie Algebra

1962 ◽  
Vol 14 ◽  
pp. 293-303 ◽  
Author(s):  
B. Noonan
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Kronecker Product ◽  
Point Of View ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Representations Of Lie Algebras ◽  
The Ideal

This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).

scholarly journals Hom-structures on semi-simple Lie algebras

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Wenjuan Xie ◽  
Quanqin Jin ◽  
Wende Liu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Jacobi Identity ◽  
Algebra Homomorphism ◽  
Closed Field ◽  
Simple Lie Algebras ◽  
3 Dimensional ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Reduction Methods

AbstractA Hom-structure on a Lie algebra (g,[,]) is a linear map σ W g σ g which satisfies the Hom-Jacobi identity: [σ(x), [y,z]] + [σ(y), [z,x]] + [σ(z),[x,y]] = 0 for all x; y; z ∈ g. A Hom-structure is referred to as multiplicative if it is also a Lie algebra homomorphism. This paper aims to determine explicitly all the Homstructures on the finite-dimensional semi-simple Lie algebras over an algebraically closed field of characteristic zero. As a Hom-structure on a Lie algebra is not necessarily a Lie algebra homomorphism, the method developed for multiplicative Hom-structures by Jin and Li in [J. Algebra 319 (2008): 1398–1408] does not work again in our case. The critical technique used in this paper, which is completely different from that in [J. Algebra 319 (2008): 1398– 1408], is that we characterize the Hom-structures on a semi-simple Lie algebra g by introducing certain reduction methods and using the software GAP. The results not only improve the earlier ones in [J. Algebra 319 (2008): 1398– 1408], but also correct an error in the conclusion for the 3-dimensional simple Lie algebra sl2. In particular, we find an interesting fact that all the Hom-structures on sl2 constitute a 6-dimensional Jordan algebra in the usual way.

scholarly journals The Square-Zero Basis of Matrix Lie Algebras

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1032
Author(s):  
Raúl Durán Díaz ◽  
Víctor Gayoso Martínez ◽  
Luis Hernández Encinas ◽  
Jaime Muñoz Masqué
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Positive Characteristic ◽  
Linear Algebraic Group ◽  
Invariant Functions ◽  
Arbitrary Characteristic ◽  
Independent Invariant ◽  
Field Of Positive Characteristic

A method is presented that allows one to compute the maximum number of functionally-independent invariant functions under the action of a linear algebraic group as long as its Lie algebra admits a basis of square-zero matrices even on a field of positive characteristic. The class of such Lie algebras is studied in the framework of the classical Lie algebras of arbitrary characteristic. Some examples and applications are also given.

Camina Lie algebras

2018 ◽  
Vol 11 (05) ◽  
pp. 1850063
Author(s):  
S. Sheikh-Mohseni ◽  
F. Saeedi
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Dimension ◽  
Proper Ideal ◽  
Closed Field ◽  
Algebraically Closed Field

Let [Formula: see text] be a Lie algebra and [Formula: see text] be a proper ideal of [Formula: see text]. Then [Formula: see text] is called a Camina pair if [Formula: see text] for all [Formula: see text]. Also, [Formula: see text] is called a Camina Lie algebra if [Formula: see text] is a Camina pair. In this paper, we give some properties of Camina Lie algebras. Moreover, we show that a nilpotent Camina Lie algebra of finite dimension over an algebraically closed field is nilpotent with nilindex at most [Formula: see text].

scholarly journals Computing with Nilpotent Orbits in Simple Lie Algebras of Exceptional Type

2008 ◽  
Vol 11 ◽  
pp. 280-297 ◽  
Author(s):  
Willem A. de Graaf
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Simple Algorithm ◽  
Adjoint Action ◽  
Closed Field ◽  
Nilpotent Orbits ◽  
Simple Algebraic Group ◽  
Algebraically Closed Field ◽  
Simple Lie Algebras

AbstractLet G be a simple algebraic group over an algebraically closed field with Lie algebra g. Then the orbits of nilpotent elements of g under the adjoint action of G have been classified. We describe a simple algorithm for finding a representative of a nilpotent orbit. We use this to compute lists of representatives of these orbits for the Lie algebras of exceptional type. Then we give two applications. The first one concerns settling a conjecture by Elashvili on the index of centralizers of nilpotent orbits, for the case where the Lie algebra is of exceptional type. The second deals with minimal dimensions of centralizers in centralizers.

scholarly journals Some Features of Rank One Real Solvable Cohomologically Rigid Lie Algebras with a Nilradical Contracting onto the Model Filiform Lie Algebra Qn

Axioms ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 10 ◽  
Author(s):  
Rutwig Campoamor-Stursberg ◽  
Francisco Oviaño García
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Maximal Torus ◽  
Real Rank ◽  
Eigenvalue Spectrum ◽  
Generic Structure ◽  
Cohomology Space ◽  
Solvable Lie Algebras ◽  
Rank One ◽  
Filiform Lie Algebra

The generic structure and some peculiarities of real rank one solvable Lie algebras possessing a maximal torus of derivations with the eigenvalue spectrum spec ( t ) = 1 , k , k + 1 , ⋯ , n + k − 3 , n + 2 k − 3 for k ≥ 2 are analyzed, with special emphasis on the resulting Lie algebras for which the second Chevalley cohomology space vanishes. From the detailed inspection of the values k ≤ 5 , some series of cohomologically rigid algebras for arbitrary values of k are determined.

scholarly journals Some Results on the Cohomology of Line Bundles on the Three Dimensional Flag Variety

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 295
Author(s):  
Muhammad Anwar
Keyword(s):  
Borel Subgroup ◽  
Three Dimensional ◽  
Linear Algebraic Group ◽  
Flag Variety ◽  
Line Bundles ◽  
Closed Field ◽  
Two Dimensional ◽  
Prime Characteristic ◽  
Simply Connected ◽  
The Given

Let k be an algebraically closed field of prime characteristic and let G be a semisimple, simply connected, linear algebraic group. It is an open problem to find the cohomology of line bundles on the flag variety G / B , where B is a Borel subgroup of G. In this paper we consider this problem in the case of G = S L 3 ( k ) and compute the cohomology for the case when ⟨ λ , α ∨ ⟩ = − p n a − 1 , ( 1 ≤ a ≤ p , n > 0 ) or ⟨ λ , α ∨ ⟩ = − p n − r , ( r ≥ 2 , n ≥ 0 ) . We also give the corresponding results for the two dimensional modules N α ( λ ) . These results will help us understand the representations of S L 3 ( k ) in the given cases.

Outer Derivations and Classical-Albert-Zassenhaus lie Algebras

1984 ◽  
Vol 36 (6) ◽  
pp. 961-972 ◽  
Author(s):  
David J. Winter
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Maximal Torus ◽  
Cartan Subalgebra ◽  
One Dimensional ◽  
Derivation Algebra ◽  
Image Position

This paper is concerned with the structure of the derivation algebra Der L of the Lie algebra L with split Cartan subalgebra H. The Fitting decompositionof Der L with respect to ad ad H leads to a decompositionwhereThis decomposition is studied in detail in Section 2, where the centralizer of ad L∞ in D0(H) is shown to bewhich is Hom(L/L2, Center L) when H is Abelian. When the root-spaces La (a nonzero) are one-dimensional, this leads to the decomposition of Der L aswhere T is any maximal torus of D0(H).

scholarly journals On Cocharacters Associated to Nilpotent Elements of Reductive Groups

2008 ◽  
Vol 190 ◽  
pp. 105-128 ◽  
Author(s):  
Russell Fowler ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Nilpotent Element ◽  
Maximal Rank ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Reductive Groups ◽  
Algebraically Closed Field ◽  
Reductive Subgroup ◽  
Good For

Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p. Assume that p is good for G. In this note we consider particular classes of connected reductive subgroups H of G and show that the cocharacters of H that are associated to a given nilpotent element e in the Lie algebra of H are precisely the cocharacters of G associated to e that take values in H. In particular, we show that this is the case provided H is a connected reductive subgroup of G of maximal rank; this answers a question posed by J. C. Jantzen.

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